Section 2.6 Reasoning With Algebra

The properties:
*The transitive property is very similar to the law of syllogism that we just used.
*The first step in two column proofs is almost always going to be given.

Proofs and Postulates

In this seccion we learned about Paragraph Proofing and Postulates. Above are the postulates we learned in class and need to be memorized. Also above is the definition that we learned, axiom or postulate.

Theorem- a statement which has been proven true.
Postulates, definitions, undefined terms and algebraic properties are used to prove theorems true.

Chapter 2 Section 4

Conjecture: an educated guess based on known imformation.

Inductive reasoning: Examining several specific situations to arrive at a conjection is called inductive reasoning.

Truth value: The truth or falsity of a statement

Law of detachment: If p=q and p is true, then q is also true

Law of syllogism: If p=q and q=r is true then p=r is true

In Chapter 2, Section 3, we learned about Conditional Statements.

A conditional statement is a statement that can be written in if-then form. When a statement is in if-then form, it is written in the form if p, then q. The part of the sentence after the word "if" is the hypothesis, and the part of the sentence after the word "then" is the conclusion.

Example 1: If you buy the television, then you will get $100 cash back.
hypothesis- You buy the television
conclusion- You get $100 cash back

Example 2: She will get a promotion if she works hard.
hypothesis- She will get a promotion
conclusion- She works hard.
hypothesis- She will get a promotion.

*There is no if-then statement in example 2. However, you can identify the hypothesis and conclusion by first identifying the hypothesis and then finding the conclusion.

Writing a Conditional Statement

First, identify the hypothesis and conclusion. Then, put them into a statement using if-then form.

Examples:

The boy is Tom's son, and he is four feet tall.
hypothesis- The boy is Tom's son
conclusion- He is four feet tall.
Conditional Statement: If he the boy is Tom's son, then he is four feet tall.

An angle with a measure of 180 degrees is a straight angle.
hypothesis- An angle has a measure of 180 degrees.
conclusion- it is a straight angle.
Conditional Statement: If an angle has a measure of 180 degrees, then is it a straight angle.

Finding the Truth Value of a Statement

The truth value of a statement is either true or false. Both parts of the statement, the hypothesis and the conclusion, must be true in order for the statement to be true.

Example:

If there is a sale, then you will buy her a pair of shoes.

a. There was a sale at the mall; you bought her a pair of shoes.
The hypothesis is true, because there was in fact a sale. The conclusion is true, because you bought her a pair of shoes, which is what was promised in the statement. Both parts are true, and therefore, the conditional statement is TRUE.

b. There was a sale at the mall; you bought her two pairs of shoes.
The hypothesis is true, because there was a sale. The conclusion is false, because you bought her two pairs of shoes, which is not what was promised. Both parts are NOT true, and therefore, the conditional statement is FALSE.

Related Conditionals

Statements based on a conditional statement are called related conditionals.

Here is a chart of the related conditonals:


Example:
Conditional: If you eat healthy food and exercise daily, then you are a healthy person.
Converse: If you are a healthy person, then you eat healthy food and exercise daily.
Inverse: If you do not eat healthy food and do not exercise daily, then you are not a healthy person.
Contrapositive: If you are not a healthy person, then do do not eat healthy food and exercise daily.

Statements with the same value are considered logically equivalent.

September 16 2009

Today in class we started logic puzzles.
We did one called five boys and five dogs. There was a situation given and then 6 clues given to help narrow down the answers.
Clues:
1) No dog's name begins with the same letter as that of his master.
2) Rover is not Bart's or Sidney's dog.
3) Spot's master and the owner of the spaniel both have names beginning with the same letter.
4) Neither Eric's dog nor Bernard's dogs is the basset, nor is Snoopy.
5) Bart's dog and the collie are not called Spot or Snoopy.
6) Ralph's dog is not a terrior

Going through these steps, some are obvious and other you have to logically think to cross other options out. Obvious ones are such as 1, 2, and 6. The others you have to think if Spots master and the owner of the spaniel..., that means that Spots master cannot own a spaniel. First go through the list and cross out the obvious and then go through a couple more times gainning more infomation as you cross more out.
Answers: Eric had a collie named Rover, Bernard had a terrior named Spot, Bart's dog was a spaniel named Fido, Sideny's dog was a basset named Bowser, Ralph's dog was a poodle named Snoopy.

In section six, we learned about polygons and the different kinds of polygons. A polygon has to have three or more segments(sides) and each side intersects to other sides and each endpoint of a side is a vertex of the polygon.
TYPES OF POLYGONS!
3 sides triangle
4 sides quadrilateral
5 sides pentagon
6 sides hexagon
7 sides heptagon
8 sides octagon
9 sides nonagon
10 sides decagon
12 sides dodecagon
If there are 11 sides or more than 12 sides, than call it an n-gon or the number and then -gon.
a shape with 11 sides= an eleven-gon
a shape with 16 sides= a sixteen-gon

We also learned about Special Polygons
Concave Polygons: A side extended goes thru the interior of the polygon.
Convex Polygons: No line that contains a side goes thru the interior of the polygon.
Equilateral Polygons: All sides are congruent
Equiangular Polygons: All interior angles are congruent
Regular Polygon: Equilateral and equiangular

Area Formulas

Area for a square: length of a side squared
Area for a rectangle: product of base and height
Area for a triangle: one-half the product of the base and height.
Perimeter of any polygon: the sum of the lengths of all the sides
Area of a circle: the product of pi and the radius squared
Circumference of a circle: 2 multiplied by pi multiplied by the radius

In Section 1-5, we learned about the different types of angle pairs and relationships. The different types of angle pairs are: adjacent angles, vertical angles, and linear pairs.


Adjacent angles are two angles that lie on the same plane, have a common vertex and common side, but no common interior points. Note that, both of the angles will always equal the total measure when added together.

Vertical angles are two nonadjacent (no common side)angles formed by two intersecting lines. The angles opposite each other will always be the same.

A Linear pair is a pair of adjacent angles with noncommon sides that are opposite rays. Both of the angles will always add up to 180 degrees.

The different types of angle relationships are: complimentary angles and supplementary angles.



Complimentary angles are two angles that have a sum of 90 degrees.

While, supplementary angles are two angles that have a sum of 180 degrees.

Types of Angles

Hey guys, I'm posting the Types of Angles picture again, because it kind of got blocked out. Sorry about that. Here it is

Section 4 (Aditya Nellore)

In section 4, we learned all about angles. An angle is made up of 2 rays, originating from the same point, which is called the vertex. Also, we learned about naming an angle. The letter that represents the vertex always has to be in the middle.(This is on the slide directly on the left). When you want to find the measure of angles, you pretty much just do it like this: m'angle'ABC=m'angle'DEF. This is also on the slide on the top left, since it's probably not very clear here. Notice you have to put an 'm' in front of the names of the angles. There was also some information about congruent angles. Congruent angles are angles that have the same measure. In other words, if you put one on top of the other, then they will fit perfectly on top of each other. We also got a list of all the types of angles and their degrees. That is in the slide on the top right. I hope this helps, guys!

In section 1.2, we covered how to be precise when measuring different types of lines. Some key terms to remember in this section were;

The reason that we have names for different types of lines in the term list is because we need to know how to measure those line precisely. We basically learned how to measure the length of lines that we were given.

Section 3

In section 3 we learned many things but two of the main terms we learned were bisect and midpoint. (Above) The midpoint is the middle point on a line. To find the midpoint you must average the x-coordinates and average the y-coordinates. We also learned how to find the distance between two points using the Pythagorean Theorem. (below) In all we
learned 3 terms or formulas that will help us complete our homework and other assignments in the future. (Midpoint, Bisect, Pythagorean Theorem)

Section One Notes (Ellie Condie)

To the left are some of the notes and terms that we discussed from section 1:

In this section we learned the basic terms of geometry and their correlation with each other. For instance a line is on a plane, and a point is on a line. We also learned other terms so that we would be familiar with different with them during the homework and class discussions. These were words such as coplanar, collinear, and congruent. Overall, we learned in this section most of the basic terms used in geometry.